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arXiv · 2010.06088

Singular Hodge theory for combinatorial geometries

Tom Braden·June Huh·Jacob P. Matherne·Nicholas Proudfoot·Botong Wang
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Abstract

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of Kazhdan-Lusztig polynomials for all matroids.

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Tom Braden, June Huh, Jacob P. Matherne, Nicholas Proudfoot, Botong Wang. 2026-06-27. Singular Hodge theory for combinatorial geometries. https://arxiv.org/abs/2010.06088

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